Extensions of Results on Rainbow Hamilton Cycles in Uniform Hypergraphs

Abstract

Let $${K_n^{(k)}}$$ K n ( k ) be the complete k-uniform hypergraph, $${k\ge3}$$ k ? 3 , and let l be an integer such that 1 ≤ l ≤ k?1 and k?l divides n. An l-overlapping Hamilton cycle in $${K_n^{(k)}}$$ K n ( k ) is a spanning subhypergraph C of $${K_n^{(k)}}$$ K n ( k ) with n/(k?l) edges and such that for some cyclic ordering of the vertices each edge of C consists of k consecutive vertices and every pair of consecutive edges in C intersects in precisely l vertices. An edge-coloring of $${K_n^{(k)}}$$ K n ( k ) is (a, r)-bounded if every subset of a vertices of $${K_n^{(k)}}$$ K n ( k ) is contained in at most r edges of the same color. In this paper, we refine recent results of the first author, Frieze and Rucinski by proving that there is a constant c = c(k, l) such that every $${(\ell, cn^{k-\ell})}$$ ( l , c n k - l ) -bounded edge-colored $${K_n^{(k)}}$$ K n ( k ) in which no color appears more that cn k-1 times contains a rainbow l-overlapping Hamilton cycle. We also show that there is a constant c? = c?(k, l) such that every (l, c?n k-l)-bounded edge-colored $${K_n^{(k)}}$$ K n ( k ) contains a properly colored l-overlapping Hamilton cycle.